On roll waves down an open inclined channel

Abstract

Flow down an open inclined channel is considered. Dressler ( Communs pure appl. Math. 2, 149-194 (1949),) using the equations of the shallow water theory augmented by the Chezy formula for drag, has shown that the uniform flow becomes unstable when the Froude number F exceeds 4, and in this case he was able to construct a one-parameter family of discontinuous periodic solutions by piecing together continuous sections of wave profile and a series of hydraulic jumps. Here the work of Dressler is extended by the inclusion of a further term that accounts for energy dissipation by shearing normal to the flow. It is shown that the inclusion of such a term does not alter the condition for stability of the uniform flow, and that when the uniform flow is unstable, a one-parameter family of quasi-steady periodic solutions exists (parametrized by the propagation speed U ), appearing as a Hopf bifurcation out of the uniform flow at the critical value U c = 1 + F -½ . After the existence of these periodic solutions has been shown, uniformly valid expansions for the periodic solutions are obtained by using the Krylov─Bogoliubov─Mitropolski averaging method, and the results are also extended to larger amplitudes by numerical integration.

Competition between kinematic and dynamic waves in floods on steep slopes

“…Two main approaches can be followed to establish the stability criteria of the plane-parallel flow: first, the hydraulic one, which finds long travelling-wave solutions for Froude numbers above 2 by means of the shallow-water equations (e.g. Jeffreys 1925;Dressler 1949;Needham & Merkin 1984;Hwang & Chang 1987;Yu & Kevorkian 1992;Chang, Demekhin & Kalaidin 2000); second, on the basis of the linearized Reynolds equations for a turbulent flow, with critical Froude numbers varying between 1.1 and 1.4 as the Reynolds number is varied over the realistic range of values 2 × 10 3 -10 5 (Demekhin, Kalaǐdin & Shapar' 2005). The effect of bottom topography on the stability of a turbulent flow over uneven surfaces was recently explored by Balmforth & Mandre (2004) following the hydraulic approach, who found that low-amplitude topography destabilizes turbulent roll waves and lowers the critical value of the Froude number required for instability.…”

Baroclinic characteristics of frictionally destabilized abyssal overflows

“…For example, Needham & Merkin (1984) examined the role of horizontal friction within the onelayer shallow-water context. Kranenburg (1992) showed that the weakly nonlinear development of a marginally unstable flow could be described by a modified Burgers equation similar to that proposed by Novik (1971).…”

Hydrodynamic Instabilities in Well-Balanced Finite Volume Schemes for Frictional Shallow Water Equations. The Kinematic Wave Case

“…We conclude by pointing out that the present work, as well as [3], could be extended by the inclusion of a further term in the momentum balance equation (4) that accounts for energy dissipation by shearing normal to the flow, as Needham and Merkin [10] did with Dressler's solution. So unrealistic extreme short wavelengths could be attenuated.…”

“…is always larger than 2 for φ > 0 and tends to the critical value of the plane-parallel flow of 2 [10] as φ → 0 (see Fig. 2(a)).…”